I could see evolving the dialogue part of an RPG to use a proof-based gameplay, where you could alternatively complete quests not by actually doing them, but by verbally convincing the NPC quest holder that you have finished the milestone. (i.e. lying?) Of course it can also be convincing them you actually did what you claim you did by using items in your inventory and some propositions.

So if you had to kill a dragon, providing a proof that you killed the dragon will give you the reward.

In RPGs, one usually has conversation-and-magic-related skills that you level up as you improve, such as intelligence, charisma, wisdom. I could imagine ideas for new benefits for the logical setup. For example, level 4 intelligence allows you to start your proof with 4 axioms. Higher level charisma could allow you to get certain proof subgoals auto-proven automatically. And higher level wisdom could simply allow for using this quest completion tactic on harder quests.

Just brainstorming here of course. It’s very easy to imagine logic used for any conversational games, maybe even a politics debate simulator where you debate your opponents, and you need to arrive at a proof of your correctness before they arrive at a refutation. You both start the race with different axioms/premises of course, which is why that’s possible.

But I also like your idea of “spells as truth” where you can unlock new spells by proving the correctness of their descriptions. The important part there is to build this on top of what I would call “Disney logic”, which is more fun than dry. I was once at a presentation by Pixar where they described how their movies use “Disney physics”, which is more entertaining than the real world. A game, unlike academics, needs to be fun first, and only then possibly useful.

Very cool idea! Here are some brainstormy thoughts.\\%
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I could see evolving the dialogue part of an RPG to use a proof-based gameplay, where you could alternatively complete quests not by actually doing them, but by verbally convincing the NPC quest holder that you have finished the milestone. (i.e. lying?) Of course it can also be convincing them you actually did what you claim you did by using items in your inventory and some propositions.\\%
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So if you had to kill a dragon, providing a proof that you killed the dragon will give you the reward.\\%
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In RPGs, one usually has conversation-and-magic-related skills that you level up as you improve, such as intelligence, charisma, wisdom. I could imagine ideas for new benefits for the logical setup. For example, level 4 intelligence allows you to start your proof with 4 axioms. Higher level charisma could allow you to get certain proof subgoals auto-proven automatically. And higher level wisdom could simply allow for using this quest completion tactic on harder quests.\\%
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Just brainstorming here of course. It's very easy to imagine logic used for any conversational games, maybe even a politics debate simulator where you debate your opponents, and you need to arrive at a proof of your correctness before they arrive at a refutation. You both start the race with different axioms/premises of course, which is why that's possible.\\%
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But I also like your idea of "spells as truth" where you can unlock new spells by proving the correctness of their descriptions. The important part there is to build this on top of what I would call "Disney logic", which is more fun than dry. I was once at a presentation by Pixar where they described how their movies use "Disney physics", which is more entertaining than the real world. A game, unlike academics, needs to be fun first, and only then possibly useful.